Nutrient shielding in clusters of cells

Abstract

Cellular nutrient consumption is influenced by both the nutrient uptake kinetics of an individual cell and the cells' spatial arrangement. Large cell clusters or colonies have inhibited growth at the cluster's center due to the shielding of nutrients by the cells closer to the surface. We develop an effective medium theory that predicts a thickness ℓ of the outer shell of cells in the cluster that receives enough nutrient to grow. The cells are treated as partially absorbing identical spherical nutrient sinks, and we identify a dimensionless parameter ν that characterizes the absorption strength of each cell. The parameter ν can vary over many orders of magnitude among different cell types, ranging from bacteria and yeast to human tissue. The thickness ℓ decreases with increasing ν, increasing cell volume fraction φ, and decreasing ambient nutrient concentration ψ(∞). The theoretical results are compared with numerical simulations and experiments. In the latter studies, colonies of budding yeast, Saccharomyces cerevisiae, are grown on glucose media and imaged under a confocal microscope. We measure the growth inside the colonies via a fluorescent protein reporter and compare the experimental and theoretical results for the thickness ℓ.

FIG. 1

(Color online) Confocal microscope images of cross-sections through the bottom of three budding yeast colonies (scale bars represent 100 μm). The red (darker shade) color is the constitutive expression of a protein, whose level is largely independent of growth rate, in all cells and the green (lighter shade) color is ribosomal protein expression, indicating growth. Colony 1: 0.5 mM glucose, 43 h after inoculation; Colony 2: 1.5 mM glucose, 47 h after inoculation; Colony 3: 4.5 mM glucose, 56 h after inoculation. Small colonies such as Colony 1 receive enough nutrients for all cells to grow. Colonies 2 and 3 are larger and growth occurs only in an outer shell of thickness ℓ (illustrated for Colony 3). The red (darkly shaded) cells in the interior are shielded from the nutrients. We assume the colonies have a spherical cap shape (see Sec. IV for a discussion of the shape and experimental details).

FIG. 2

(Color online) A plot of the nutrient current I_cell (solid red line) into a cell as a function of the ambient nutrient concentration ψ_∞, which follows Michaelis-Menten kinetics. The maximum current I_max and the kinetics parameter K_m are also shown.

FIG. 3

(Color online) A radially symmetric model of nutrient uptake in a single cell. We specify a steady concentration ψ_∞ far away from the cell. The cell is centered at the origin and has radius a. The steady state concentration profile ψ₀(r) (black solid line) is calculated for a rectangular potential barrier (red dashed line) of height u₀ (designed to model the complex uptake dynamics of the cell wall) and a perfectly absorbing nutrient sink at r = a′. The concentration profile approaches ψ_∞ at large r and exhibits a jump discontinuity at r = a. The nutrient currents are constrained to be continuous.

FIG. 4

(Color online) Schematic of the effective medium approximation, which replaces N orange (gray) cells in a spherical cluster with radius b (enclosed by the dashed lines) in (a) by a homogeneous, attenuating medium in (b) shown in dark and light shades of green. The lightly shaded green rim of width ℓ illustrates the section of the cluster receiving enough nutrients to grow. ψ_∞ is the limiting nutrient concentration in the variably shaded red region, outside the cluster. To analyze this problem, we use the coordinate system in (c), with an origin at $\mathcal{O}$, cell centers located at ${\left\{{\mathbf{R}}_i^0\right\}}_{i=1}^N$, and vectors r_i pointing from the cell center to the surface of the cell.

FIG. 5

(Color online) Plot of the thickness ℓ of actively growing cells in a spherical cluster of size b as a function of the nutrient bath concentration ψ_∞ for various values of the penetration depth ξ.

FIG. 6

(Color online) Plot of the screening length ξd in the small ϕ limit as a function of ν for various values of ϕ on a log-log scale. ξ crosses over to the perfectly absorbing limit at ν ~ 1. The arrows Y and B denote, respectively, typical values of ν for baker's yeast and bacteria.

FIG. 7

(Color online) Plot of the screening length ξ calculated with the effective medium theory divided by the dilute limit result ξ_d as a function of the volume fraction ϕ for various values of ν. The low density result becomes increasingly inaccurate at larger packing fractions.

FIG. 8

(Color online) Plot of the diffusion coefficient D in the cell cluster calculated with the effective medium theory divided by the bare diffusion coefficient D₀ in the absence of the cluster as a function of the volume fraction ϕ for various values of ν.

FIG. 9

(Color online) A simulated cluster of cells with ν = 1 and ψ_∞ = 1. The color (shading) indicates the local nutrient concentration Ψ(r) near the cell surfaces. In this case, the 200 cells with radius a were confined to a cluster of radius b ≈ 7.45a, yielding a volume fraction of ϕ ≈ 0.48. The radius of the large bounding sphere on which the nutrient concentration was fixed at ψ_∞ was about 56a ≈ 7.5b. To better simulate the fixed concentration ψ_∞ infinitely far from the cluster, we used COMSOL's “infinite element” option with spherical symmetry between radii 28a and 56a (see Sec. III A).

FIG. 10

(Color online) The radially averaged concentration rΨ̵(r)/aψ_∞ (symbols) and the effective medium result rψ(r)/aψ_∞ (lines) for a cluster of 353 cells of radius a (with cluster radius b ≈ 8.15a) as a function of δ = (b − r)/a on a log-linear plot. The concentrations are rescaled to highlight the exponential decay of the concentration into the bulk of the cluster. Cells occupy the region 0 < δ < 8.15 with the center at δ ≈ 8.15, as indicated by the dashed lines. The cell packing fraction is ϕ ≈ 0.63. The bounding sphere has a radius of 50a, with “infinite elements” inserted at distances between 25a and 50a (see Sec. III A).

FIG. 11

(Color online) The radially averaged concentration Ψ̵(r)/ψ_∞ (symbols) and the effective medium solution ψ(r)/ψ_∞ (lines) as a function of δ = (b − r)/a for the same 353 cell cluster described in Fig. 10. The dotted lines indicate the cluster edge (δ = 0) and center (δ ≈ 8.15).

FIG. 12

(Color online) Exponential decay of the rescaled nutrient concentration in the effective medium theory (lines) and in a simulation (symbols) for a more dilute 85 cell cluster with b = 7a and various ν. The cell packing fraction is ϕ ≈ 0.24. The cluster, shown above the graph, now occupies the region between the dotted lines, 0 < δ < 7.

FIG. 13

(Color online) (a) Confocal microscopy setup. A single yeast cell was inoculated into a well of a glass-bottomed 96-well plate containing 200 μL of yeast media and allowed to grow into a colony. The bottom of the colony was imaged using a spinning-disc confocal inverted microscope. (b) Cells were grown in shaken media, pipetted into an empty well, and imaged immediately (before they could divide and form colonies) with the setup shown in (a). At least 250 cells at each concentration were imaged, and the error bars are the 95% confidence interval using the Student's t-test. Relative fluorescence intensity in (b) is defined as ribosomal protein expression divided by constitutive expression. Cells in 0.167 mM glucose continue to divide, but at less than half the growth rate of cells in 0.5 mM glucose.

FIG. 14

(Color online) Relative fluorescence intensity (ribosomal protein expression divided by constitutive expression) as a function of distance d = b−r from the edge of the colony. Here b is the colony radius and r is the observation position. Each point is the average fluorescence of an 8 μm thick ring whose edge is located an equal distance from the edge of the colony. The last point on each connected set of lines represents the radius of the colony and larger radii reflect longer periods of incubation. Each line is one colony. Colonies were grown in 0.5 mM glucose (top), 1.5 mM glucose (middle), and 4.5 mM glucose (bottom). Profiles from the three colonies shown in Fig. 1 are shown as orange (gray) triangles.

FIG. 15

(Color online) Penetration depth ℓ as a function of colony radius b. Penetration depth in the experiment (symbols) is the distance from the edge of the colony at which relative fluorescence drops below 0.4. The theoretical results (solid lines) are calculated by numerically solving ψ(r = b − ℓ) = ψ_min (Eq. 21) for ℓ, where we assume that ψ_min ≈ 0.25 mM local glucose concentration corresponds to a 0.4 relative fluorescence level. As discussed in the text, this level is slightly higher than the level at which the cells stop growing and expressing the ribosomal protein. We also use a cell radius of a = 2 μm, a cell packing fraction ϕ = 0.56, and ν = 6 × 10⁻⁴.